### Abstract

We study the Hecke algebra (Formula presented.) over an arbitrary field (Formula presented.) of a Coxeter system (W, S) with independent parameters (Formula presented.) for all generators. This algebra always has a spanning set indexed by the Coxeter group W, which is indeed a basis if and only if every pair of generators joined by an odd edge in the Coxeter diagram receives the same parameter. In general, the dimension of (Formula presented.) could be as small as 1. We construct a basis for (Formula presented.) when (W, S) is simply laced. We also characterize when (Formula presented.) is commutative, which happens only if the Coxeter diagram of (W, S) is simply laced and bipartite. In particular, for type A, we obtain a tower of semisimple commutative algebras whose dimensions are the Fibonacci numbers. We show that the representation theory of these algebras has some features in analogy/connection with the representation theory of the symmetric groups and the 0-Hecke algebras.

Original language | English (US) |
---|---|

Pages (from-to) | 521-551 |

Number of pages | 31 |

Journal | Journal of Algebraic Combinatorics |

Volume | 43 |

Issue number | 3 |

DOIs | |

State | Published - May 1 2016 |

### Fingerprint

### Keywords

- Fibonacci number
- Grothendieck group
- Hecke algebra
- Independent parameters
- Independent set

### ASJC Scopus subject areas

- Algebra and Number Theory
- Discrete Mathematics and Combinatorics

### Cite this

**Hecke algebras with independent parameters.** / Huang, Jia.

Research output: Contribution to journal › Article

*Journal of Algebraic Combinatorics*, vol. 43, no. 3, pp. 521-551. https://doi.org/10.1007/s10801-015-0645-7

}

TY - JOUR

T1 - Hecke algebras with independent parameters

AU - Huang, Jia

PY - 2016/5/1

Y1 - 2016/5/1

N2 - We study the Hecke algebra (Formula presented.) over an arbitrary field (Formula presented.) of a Coxeter system (W, S) with independent parameters (Formula presented.) for all generators. This algebra always has a spanning set indexed by the Coxeter group W, which is indeed a basis if and only if every pair of generators joined by an odd edge in the Coxeter diagram receives the same parameter. In general, the dimension of (Formula presented.) could be as small as 1. We construct a basis for (Formula presented.) when (W, S) is simply laced. We also characterize when (Formula presented.) is commutative, which happens only if the Coxeter diagram of (W, S) is simply laced and bipartite. In particular, for type A, we obtain a tower of semisimple commutative algebras whose dimensions are the Fibonacci numbers. We show that the representation theory of these algebras has some features in analogy/connection with the representation theory of the symmetric groups and the 0-Hecke algebras.

AB - We study the Hecke algebra (Formula presented.) over an arbitrary field (Formula presented.) of a Coxeter system (W, S) with independent parameters (Formula presented.) for all generators. This algebra always has a spanning set indexed by the Coxeter group W, which is indeed a basis if and only if every pair of generators joined by an odd edge in the Coxeter diagram receives the same parameter. In general, the dimension of (Formula presented.) could be as small as 1. We construct a basis for (Formula presented.) when (W, S) is simply laced. We also characterize when (Formula presented.) is commutative, which happens only if the Coxeter diagram of (W, S) is simply laced and bipartite. In particular, for type A, we obtain a tower of semisimple commutative algebras whose dimensions are the Fibonacci numbers. We show that the representation theory of these algebras has some features in analogy/connection with the representation theory of the symmetric groups and the 0-Hecke algebras.

KW - Fibonacci number

KW - Grothendieck group

KW - Hecke algebra

KW - Independent parameters

KW - Independent set

UR - http://www.scopus.com/inward/record.url?scp=84946135164&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84946135164&partnerID=8YFLogxK

U2 - 10.1007/s10801-015-0645-7

DO - 10.1007/s10801-015-0645-7

M3 - Article

AN - SCOPUS:84946135164

VL - 43

SP - 521

EP - 551

JO - Journal of Algebraic Combinatorics

JF - Journal of Algebraic Combinatorics

SN - 0925-9899

IS - 3

ER -